Forty Years of Meshless Radial Basis Function Applications to Integral and Partial Differential Equations

Abstract

This review paper highlights the versatility of meshless radial basis functions (RBFs) for the numerical solution of linear and nonlinear integral equations (IEs), elliptic, hyperbolic, and parabolic partial differential equations (PDEs), and integro-partial differential equations (IPDEs). In addition, we show that classical well-ordered discretizations can be a serious hindrance to overcoming the curse of dimensionality, whereas randomly generated points do not exhibit the same exponential growth in complexity. Finally, we briefly discuss how very high-dimensional problems may be addressed on emerging computing architectures (including quantum devices) using neural networks and artificial intelligence.